Cantor's Intersection Theorem

Cantor's intersection theorem, also known as Cantor's nested intervals theorem, refers to a pair of closely related results in general topology and real analysis, both named after Georg Cantor. The theorems concern what happens when an infinite sequence of nonempty compact sets is nested inside one another, each contained in the one before it, and they establish that under these conditions the intersection of the entire sequence is guaranteed to be nonempty. This guarantee depends specifically on the sets being compact and nested in a decreasing chain, and it is a foundational tool used throughout analysis and topology wherever an infinite nested sequence of sets needs to be shown to have a common point.